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Lecture 6: Model-Free Control
Lecture 6: Model-Free Control
Hado van Hasselt
Lecture 6: Model-Free Control
Outline
1 Introduction
2 Monte-Carlo Control
3 On-Policy Temporal-Difference Learning
4 Off-Policy Learning
5 Summary
Lecture 6: Model-Free Control
Introduction
Outline
1 Introduction
2 Monte-Carlo Control
3 On-Policy Temporal-Difference Learning
4 Off-Policy Learning
5 Summary
Lecture 6: Model-Free Control
Introduction
Model-Free Reinforcement Learning
Last lecture:
Model-free predictionEstimate the value function of an unknown MDP
This lecture:
Model-free controlOptimise the value function of an unknown MDP
Lecture 6: Model-Free Control
Introduction
Model-Free Reinforcement Learning
Last lecture:
Model-free prediction
Estimate the value function of an unknown MDP
This lecture:
Model-free controlOptimise the value function of an unknown MDP
Lecture 6: Model-Free Control
Introduction
Model-Free Reinforcement Learning
Last lecture:
Model-free predictionEstimate the value function of an unknown MDP
This lecture:
Model-free controlOptimise the value function of an unknown MDP
Lecture 6: Model-Free Control
Introduction
Model-Free Reinforcement Learning
Last lecture:
Model-free predictionEstimate the value function of an unknown MDP
This lecture:
Model-free controlOptimise the value function of an unknown MDP
Lecture 6: Model-Free Control
Introduction
Model-Free Reinforcement Learning
Last lecture:
Model-free predictionEstimate the value function of an unknown MDP
This lecture:
Model-free control
Optimise the value function of an unknown MDP
Lecture 6: Model-Free Control
Introduction
Model-Free Reinforcement Learning
Last lecture:
Model-free predictionEstimate the value function of an unknown MDP
This lecture:
Model-free controlOptimise the value function of an unknown MDP
Lecture 6: Model-Free Control
Introduction
Uses of Model-Free Control
Some example problems that can be modelled as MDPs
Elevator
Parallel Parking
Ship Steering
Bioreactor
Helicopter
Aeroplane Logistics
Robocup Soccer
Portfolio management
Protein Folding
Robot walking
Atari video games
Game of Go
For most of these problems, either:
MDP model is unknown, but experience can be sampled
MDP model is known, but is too big to use, except by samples
Model-free control can solve these problems
Lecture 6: Model-Free Control
Introduction
Uses of Model-Free Control
Some example problems that can be modelled as MDPs
Elevator
Parallel Parking
Ship Steering
Bioreactor
Helicopter
Aeroplane Logistics
Robocup Soccer
Portfolio management
Protein Folding
Robot walking
Atari video games
Game of Go
For most of these problems, either:
MDP model is unknown, but experience can be sampled
MDP model is known, but is too big to use, except by samples
Model-free control can solve these problems
Lecture 6: Model-Free Control
Introduction
Uses of Model-Free Control
Some example problems that can be modelled as MDPs
Elevator
Parallel Parking
Ship Steering
Bioreactor
Helicopter
Aeroplane Logistics
Robocup Soccer
Portfolio management
Protein Folding
Robot walking
Atari video games
Game of Go
For most of these problems, either:
MDP model is unknown, but experience can be sampled
MDP model is known, but is too big to use, except by samples
Model-free control can solve these problems
Lecture 6: Model-Free Control
Introduction
Uses of Model-Free Control
Some example problems that can be modelled as MDPs
Elevator
Parallel Parking
Ship Steering
Bioreactor
Helicopter
Aeroplane Logistics
Robocup Soccer
Portfolio management
Protein Folding
Robot walking
Atari video games
Game of Go
For most of these problems, either:
MDP model is unknown, but experience can be sampled
MDP model is known, but is too big to use, except by samples
Model-free control can solve these problems
Lecture 6: Model-Free Control
Monte-Carlo Control
Outline
1 Introduction
2 Monte-Carlo Control
3 On-Policy Temporal-Difference Learning
4 Off-Policy Learning
5 Summary
Lecture 6: Model-Free Control
Monte-Carlo Control
Generalized Policy Iteration
Generalized Policy Iteration (Refresher)
Policy evaluation Estimate V π
e.g. Iterative policy evaluation
Policy improvement Generate π′ ≥ πe.g. Greedy policy improvement
Lecture 6: Model-Free Control
Monte-Carlo Control
Generalized Policy Iteration
Monte Carlo
Recall, Monte Carlo estimate from state St is
Gt = Rt+1 + γRt+2 + γ2Rt+3 + . . .
E[Gt ] = V π
So, we can multiple estimates to get V π
Lecture 6: Model-Free Control
Monte-Carlo Control
Generalized Policy Iteration
Monte Carlo
Recall, Monte Carlo estimate from state St is
Gt = Rt+1 + γRt+2 + γ2Rt+3 + . . .
E[Gt ] = V π
So, we can multiple estimates to get V π
Lecture 6: Model-Free Control
Monte-Carlo Control
Generalized Policy Iteration
Monte Carlo
Recall, Monte Carlo estimate from state St is
Gt = Rt+1 + γRt+2 + γ2Rt+3 + . . .
E[Gt ] = V π
So, we can multiple estimates to get V π
Lecture 6: Model-Free Control
Monte-Carlo Control
Generalized Policy Iteration
Policy Iteration With Monte-Carlo Evaluation
Policy evaluation Monte-Carlo policy evaluation, V = V π?
Policy improvement Greedy policy improvement?
Lecture 6: Model-Free Control
Monte-Carlo Control
Generalized Policy Iteration
Model-Free Policy Iteration Using Action-Value Function
Greedy policy improvement over V (s) requires model of MDP
π′(s) = argmaxa∈A
RAs + PA
ss′V (s ′)
Greedy policy improvement over Q(s, a) is model-free
π′(s) = argmaxa∈A
Q(s, a)
Lecture 6: Model-Free Control
Monte-Carlo Control
Generalized Policy Iteration
Model-Free Policy Iteration Using Action-Value Function
Greedy policy improvement over V (s) requires model of MDP
π′(s) = argmaxa∈A
RAs + PA
ss′V (s ′)
Greedy policy improvement over Q(s, a) is model-free
π′(s) = argmaxa∈A
Q(s, a)
Lecture 6: Model-Free Control
Monte-Carlo Control
Generalized Policy Iteration
Generalised Policy Iteration with Action-Value Function
Starting Q, π
π = greedy(Q)
Q = Q π
Q*, π*
Policy evaluation Monte-Carlo policy evaluation, Q = Qπ
Policy improvement Greedy policy improvement?
Lecture 6: Model-Free Control
Monte-Carlo Control
Generalized Policy Iteration
ε-Greedy Policy Improvement
Theorem
For any ε-greedy policy π, the ε-greedy policy π′ with respect toQπ is an improvement, V π′
(s) ≥ V π(s)
Qπ(s, π′(s)) =∑
a∈Aπ′(s, a)Qπ(s, a)
= ε/m∑
a∈AQπ(s, a) + (1− ε) max
a∈AQπ(s, a)
≥ ε/m∑
a∈AQπ(s, a) + (1− ε)
∑
a∈A
π(s, a)− ε/m1− ε Qπ(s, a)
=∑
a∈Aπ(s, a)Qπ(s, a) = V π(s)
Therefore from policy improvement theorem, V π′(s) ≥ V π(s)
Lecture 6: Model-Free Control
Monte-Carlo Control
Generalized Policy Iteration
ε-Greedy Policy Improvement
Theorem
For any ε-greedy policy π, the ε-greedy policy π′ with respect toQπ is an improvement, V π′
(s) ≥ V π(s)
Qπ(s, π′(s)) =∑
a∈Aπ′(s, a)Qπ(s, a)
= ε/m∑
a∈AQπ(s, a) + (1− ε) max
a∈AQπ(s, a)
≥ ε/m∑
a∈AQπ(s, a) + (1− ε)
∑
a∈A
π(s, a)− ε/m1− ε Qπ(s, a)
=∑
a∈Aπ(s, a)Qπ(s, a) = V π(s)
Therefore from policy improvement theorem, V π′(s) ≥ V π(s)
Lecture 6: Model-Free Control
Monte-Carlo Control
Generalized Policy Iteration
Monte-Carlo Policy Iteration
Starting Q, π
π = ε-greedy(Q)
Q = Q π
Q*, π*
Policy evaluation Monte-Carlo policy evaluation, Q = Qπ
Policy improvement ε-greedy policy improvement
Here π∗ is best ε-greedy policy
Lecture 6: Model-Free Control
Monte-Carlo Control
Generalized Policy Iteration
Monte-Carlo Generalized Policy Iteration
Starting Q
π = ε-greedy(Q)
Q = Q π
Q*, π*
Every episode:
Policy evaluation Monte-Carlo policy evaluation, Q ≈ Qπ
Policy improvement ε-greedy policy improvement
Lecture 6: Model-Free Control
Monte-Carlo Control
GLIE
GLIE
Definition
Greedy in the Limit with Infinite Exploration (GLIE)
All state-action pairs are explored infinitely many times,
limk→∞
Nk(s, a) =∞
The policy converges on a greedy policy,
limk→∞
πk(s, a) = 1(a = argmaxa′∈A
Qk(s, a′))
For example, ε-greedy is GLIE if ε reduces to zero at εk = 1k
Lecture 6: Model-Free Control
Monte-Carlo Control
GLIE
GLIE
Definition
Greedy in the Limit with Infinite Exploration (GLIE)
All state-action pairs are explored infinitely many times,
limk→∞
Nk(s, a) =∞
The policy converges on a greedy policy,
limk→∞
πk(s, a) = 1(a = argmaxa′∈A
Qk(s, a′))
For example, ε-greedy is GLIE if ε reduces to zero at εk = 1k
Lecture 6: Model-Free Control
Monte-Carlo Control
GLIE
GLIE
Definition
Greedy in the Limit with Infinite Exploration (GLIE)
All state-action pairs are explored infinitely many times,
limk→∞
Nk(s, a) =∞
The policy converges on a greedy policy,
limk→∞
πk(s, a) = 1(a = argmaxa′∈A
Qk(s, a′))
For example, ε-greedy is GLIE if ε reduces to zero at εk = 1k
Lecture 6: Model-Free Control
Monte-Carlo Control
GLIE
GLIE Every-Visit Monte-Carlo Control
Sample kth episode using π: {S1,A1,R2, ...,ST} ∼ π
For each state St and action At in the episode,
N(St ,At)← N(St ,At) + 1
Q(St ,At)← Q(St ,At) +1
N(St ,At)(Gt − Q(St ,At))
Improve policy based on new action-value function
ε← 1/k
π ← ε-greedy(Q)
Theorem
GLIE Monte-Carlo control converges to the optimal action-valuefunction, Q(s, a)→ Q∗(s, a)
Lecture 6: Model-Free Control
Monte-Carlo Control
GLIE
GLIE Every-Visit Monte-Carlo Control
Sample kth episode using π: {S1,A1,R2, ...,ST} ∼ πFor each state St and action At in the episode,
N(St ,At)← N(St ,At) + 1
Q(St ,At)← Q(St ,At) +1
N(St ,At)(Gt − Q(St ,At))
Improve policy based on new action-value function
ε← 1/k
π ← ε-greedy(Q)
Theorem
GLIE Monte-Carlo control converges to the optimal action-valuefunction, Q(s, a)→ Q∗(s, a)
Lecture 6: Model-Free Control
Monte-Carlo Control
GLIE
GLIE Every-Visit Monte-Carlo Control
Sample kth episode using π: {S1,A1,R2, ...,ST} ∼ πFor each state St and action At in the episode,
N(St ,At)← N(St ,At) + 1
Q(St ,At)← Q(St ,At) +1
N(St ,At)(Gt − Q(St ,At))
Improve policy based on new action-value function
ε← 1/k
π ← ε-greedy(Q)
Theorem
GLIE Monte-Carlo control converges to the optimal action-valuefunction, Q(s, a)→ Q∗(s, a)
Lecture 6: Model-Free Control
Monte-Carlo Control
GLIE
GLIE Every-Visit Monte-Carlo Control
Sample kth episode using π: {S1,A1,R2, ...,ST} ∼ πFor each state St and action At in the episode,
N(St ,At)← N(St ,At) + 1
Q(St ,At)← Q(St ,At) +1
N(St ,At)(Gt − Q(St ,At))
Improve policy based on new action-value function
ε← 1/k
π ← ε-greedy(Q)
Theorem
GLIE Monte-Carlo control converges to the optimal action-valuefunction, Q(s, a)→ Q∗(s, a)
Lecture 6: Model-Free Control
Monte-Carlo Control
GLIE
GLIE Every-Visit Monte-Carlo Control
Sample kth episode using π: {S1,A1,R2, ...,ST} ∼ πFor each state St and action At in the episode,
N(St ,At)← N(St ,At) + 1
Q(St ,At)← Q(St ,At) +1
N(St ,At)(Gt − Q(St ,At))
Improve policy based on new action-value function
ε← 1/k
π ← ε-greedy(Q)
Theorem
GLIE Monte-Carlo control converges to the optimal action-valuefunction, Q(s, a)→ Q∗(s, a)
Lecture 6: Model-Free Control
Monte-Carlo Control
GLIE
GLIE Every-Visit Monte-Carlo Control
N(St ,At)← N(St ,At) + 1
Q(St ,At)← Q(St ,At) +1
N(St ,At)(Gt − Q(St ,At))
ε← 1/k
π ← ε-greedy(Q)
Any practical issues with this algorithm?
Any ways to improve, in practice?
Is 1/N the best step size?
Is 1/k enough exploration?
Lecture 6: Model-Free Control
Monte-Carlo Control
GLIE
GLIE Every-Visit Monte-Carlo Control
N(St ,At)← N(St ,At) + 1
Q(St ,At)← Q(St ,At) +1
N(St ,At)(Gt − Q(St ,At))
ε← 1/k
π ← ε-greedy(Q)
Any practical issues with this algorithm?
Any ways to improve, in practice?
Is 1/N the best step size?
Is 1/k enough exploration?
Lecture 6: Model-Free Control
Monte-Carlo Control
GLIE
GLIE Every-Visit Monte-Carlo Control
N(St ,At)← N(St ,At) + 1
Q(St ,At)← Q(St ,At) +1
N(St ,At)(Gt − Q(St ,At))
ε← 1/k
π ← ε-greedy(Q)
Any practical issues with this algorithm?
Any ways to improve, in practice?
Is 1/N the best step size?
Is 1/k enough exploration?
Lecture 6: Model-Free Control
Monte-Carlo Control
GLIE
GLIE Every-Visit Monte-Carlo Control
N(St ,At)← N(St ,At) + 1
Q(St ,At)← Q(St ,At) +1
N(St ,At)(Gt − Q(St ,At))
ε← 1/k
π ← ε-greedy(Q)
Any practical issues with this algorithm?
Any ways to improve, in practice?
Is 1/N the best step size?
Is 1/k enough exploration?
Lecture 6: Model-Free Control
Monte-Carlo Control
Blackjack Example
Back to the Blackjack Example
Lecture 6: Model-Free Control
Monte-Carlo Control
Blackjack Example
Monte-Carlo Control in Blackjack
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Outline
1 Introduction
2 Monte-Carlo Control
3 On-Policy Temporal-Difference Learning
4 Off-Policy Learning
5 Summary
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
MC vs. TD Control
Temporal-difference (TD) learning has several advantagesover Monte-Carlo (MC)
Lower varianceOnlineCan learn from incomplete sequences
Natural idea: use TD instead of MC for control
Apply TD to Q(s, a)Use ε-greedy policy improvementUpdate every time-step
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
MC vs. TD Control
Temporal-difference (TD) learning has several advantagesover Monte-Carlo (MC)
Lower variance
OnlineCan learn from incomplete sequences
Natural idea: use TD instead of MC for control
Apply TD to Q(s, a)Use ε-greedy policy improvementUpdate every time-step
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
MC vs. TD Control
Temporal-difference (TD) learning has several advantagesover Monte-Carlo (MC)
Lower varianceOnline
Can learn from incomplete sequences
Natural idea: use TD instead of MC for control
Apply TD to Q(s, a)Use ε-greedy policy improvementUpdate every time-step
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
MC vs. TD Control
Temporal-difference (TD) learning has several advantagesover Monte-Carlo (MC)
Lower varianceOnlineCan learn from incomplete sequences
Natural idea: use TD instead of MC for control
Apply TD to Q(s, a)Use ε-greedy policy improvementUpdate every time-step
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
MC vs. TD Control
Temporal-difference (TD) learning has several advantagesover Monte-Carlo (MC)
Lower varianceOnlineCan learn from incomplete sequences
Natural idea: use TD instead of MC for control
Apply TD to Q(s, a)Use ε-greedy policy improvementUpdate every time-step
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
MC vs. TD Control
Temporal-difference (TD) learning has several advantagesover Monte-Carlo (MC)
Lower varianceOnlineCan learn from incomplete sequences
Natural idea: use TD instead of MC for control
Apply TD to Q(s, a)
Use ε-greedy policy improvementUpdate every time-step
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
MC vs. TD Control
Temporal-difference (TD) learning has several advantagesover Monte-Carlo (MC)
Lower varianceOnlineCan learn from incomplete sequences
Natural idea: use TD instead of MC for control
Apply TD to Q(s, a)Use ε-greedy policy improvement
Update every time-step
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
MC vs. TD Control
Temporal-difference (TD) learning has several advantagesover Monte-Carlo (MC)
Lower varianceOnlineCan learn from incomplete sequences
Natural idea: use TD instead of MC for control
Apply TD to Q(s, a)Use ε-greedy policy improvementUpdate every time-step
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Updating Action-Value Functions with Sarsa
s,a
r
a'
s'
Q(s, a)← Q(s, a) + α(R + γQ(s ′, a′)− Q(s, a)
)
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Sarsa
Starting Q
π = ε-greedy(Q)
Q = Q π
Q*, π*
Every time-step:
Policy evaluation Sarsa, Q ≈ Qπ
Policy improvement ε-greedy policy improvement
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Sarsa
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Convergence of Sarsa
Theorem
Sarsa converges to the optimal action-value function,Q(s, a)→ Q∗(s, a), under the following conditions:
GLIE sequence of policies πt(s, a)
Robbins-Monro sequence of step-sizes αt
∞∑
t=1
αt =∞
∞∑
t=1
α2t <∞
E.g., αt = 1/t or αt = 1/tω with ω ∈ (0.5, 1).
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Convergence of Sarsa
Theorem
Sarsa converges to the optimal action-value function,Q(s, a)→ Q∗(s, a), under the following conditions:
GLIE sequence of policies πt(s, a)
Robbins-Monro sequence of step-sizes αt
∞∑
t=1
αt =∞
∞∑
t=1
α2t <∞
E.g., αt = 1/t or αt = 1/tω with ω ∈ (0.5, 1).
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Windy Gridworld Example
Reward = -1 per time-step until reaching goal
Undiscounted
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Windy Gridworld Example
Reward = -1 per time-step until reaching goal
Undiscounted
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Sarsa on the Windy Gridworld
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
n-Step Sarsa
Consider the following n-step returns for n = 1, 2, . . . ,∞:
n = 1 Sarsa(0) G(1)t = Rt+1 + γQ(St+1)
n = 2 G(2)t = Rt+1 + γRt+2 + γ2Q(St+2)
......
n =∞ MC G(∞)t = Rt+1 + γRt+2 + ...+ γT−1RT
Define the n-step Q-return
G(n)t = Rt+1 + γRt+2 + ...+ γn−1Rt+n + γnQ(St+n)
n-step Sarsa updates Q(s, a) towards the n-step Q-return
Q(St ,At)← Q(St ,At) + α(G
(n)t − Q(St ,At)
)
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
n-Step Sarsa
Consider the following n-step returns for n = 1, 2, . . . ,∞:
n = 1 Sarsa(0) G(1)t = Rt+1 + γQ(St+1)
n = 2 G(2)t = Rt+1 + γRt+2 + γ2Q(St+2)
......
n =∞ MC G(∞)t = Rt+1 + γRt+2 + ...+ γT−1RT
Define the n-step Q-return
G(n)t = Rt+1 + γRt+2 + ...+ γn−1Rt+n + γnQ(St+n)
n-step Sarsa updates Q(s, a) towards the n-step Q-return
Q(St ,At)← Q(St ,At) + α(G
(n)t − Q(St ,At)
)
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
n-Step Sarsa
Consider the following n-step returns for n = 1, 2, . . . ,∞:
n = 1 Sarsa(0) G(1)t = Rt+1 + γQ(St+1)
n = 2 G(2)t = Rt+1 + γRt+2 + γ2Q(St+2)
......
n =∞ MC G(∞)t = Rt+1 + γRt+2 + ...+ γT−1RT
Define the n-step Q-return
G(n)t = Rt+1 + γRt+2 + ...+ γn−1Rt+n + γnQ(St+n)
n-step Sarsa updates Q(s, a) towards the n-step Q-return
Q(St ,At)← Q(St ,At) + α(G
(n)t − Q(St ,At)
)
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Forward View Sarsa(λ)
The Gλ return combines all n-step
Q-returns G(n)t
Using weight (1− λ)λn−1
Gλt = (1− λ)
∞∑
n=1
λn−1G(n)t
Forward-view Sarsa(λ)
Q(St ,At)← Q(St ,At) + α(Gλt − Q(St ,At)
)
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Forward View Sarsa(λ)
The Gλ return combines all n-step
Q-returns G(n)t
Using weight (1− λ)λn−1
Gλt = (1− λ)
∞∑
n=1
λn−1G(n)t
Forward-view Sarsa(λ)
Q(St ,At)← Q(St ,At) + α(Gλt − Q(St ,At)
)
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Forward View Sarsa(λ)
The Gλ return combines all n-step
Q-returns G(n)t
Using weight (1− λ)λn−1
Gλt = (1− λ)
∞∑
n=1
λn−1G(n)t
Forward-view Sarsa(λ)
Q(St ,At)← Q(St ,At) + α(Gλt − Q(St ,At)
)
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Forward View Sarsa(λ)
Written recursively:
Gλt = Rt+1 + γ(1− λ)Qt(St+1,At+1) + γλGλ
t+1
Weight 1 on Rt+1
Weight γ(1− λ) on Qt(St+1,At+1)
Weight γλ on Rt+2
Weight γ2λ(1− λ) on Qt(St+2,At+2)
Weight γ2λ2 on Rt+3
Weight γ3λ2(1− λ) on Qt(St+3,At+3)
...
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Forward View Sarsa(λ)
Written recursively:
Gλt = Rt+1 + γ(1− λ)Qt(St+1,At+1) + γλGλ
t+1
Weight 1 on Rt+1
Weight γ(1− λ) on Qt(St+1,At+1)
Weight γλ on Rt+2
Weight γ2λ(1− λ) on Qt(St+2,At+2)
Weight γ2λ2 on Rt+3
Weight γ3λ2(1− λ) on Qt(St+3,At+3)
...
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Forward View Sarsa(λ)
Written recursively:
Gλt = Rt+1 + γ(1− λ)Qt(St+1,At+1) + γλGλ
t+1
Weight 1 on Rt+1
Weight γ(1− λ) on Qt(St+1,At+1)
Weight γλ on Rt+2
Weight γ2λ(1− λ) on Qt(St+2,At+2)
Weight γ2λ2 on Rt+3
Weight γ3λ2(1− λ) on Qt(St+3,At+3)
...
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Forward View Sarsa(λ)
Written recursively:
Gλt = Rt+1 + γ(1− λ)Qt(St+1,At+1) + γλGλ
t+1
Weight 1 on Rt+1
Weight γ(1− λ) on Qt(St+1,At+1)
Weight γλ on Rt+2
Weight γ2λ(1− λ) on Qt(St+2,At+2)
Weight γ2λ2 on Rt+3
Weight γ3λ2(1− λ) on Qt(St+3,At+3)
...
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Forward View Sarsa(λ)
Written recursively:
Gλt = Rt+1 + γ(1− λ)Qt(St+1,At+1) + γλGλ
t+1
Weight 1 on Rt+1
Weight γ(1− λ) on Qt(St+1,At+1)
Weight γλ on Rt+2
Weight γ2λ(1− λ) on Qt(St+2,At+2)
Weight γ2λ2 on Rt+3
Weight γ3λ2(1− λ) on Qt(St+3,At+3)
...
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Forward View Sarsa(λ)
Written recursively:
Gλt = Rt+1 + γ(1− λ)Qt(St+1,At+1) + γλGλ
t+1
Weight 1 on Rt+1
Weight γ(1− λ) on Qt(St+1,At+1)
Weight γλ on Rt+2
Weight γ2λ(1− λ) on Qt(St+2,At+2)
Weight γ2λ2 on Rt+3
Weight γ3λ2(1− λ) on Qt(St+3,At+3)
...
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Forward View Sarsa(λ)
Written recursively:
Gλt = Rt+1 + γ(1− λ)Qt(St+1,At+1) + γλGλ
t+1
Weight 1 on Rt+1
Weight γ(1− λ) on Qt(St+1,At+1)
Weight γλ on Rt+2
Weight γ2λ(1− λ) on Qt(St+2,At+2)
Weight γ2λ2 on Rt+3
Weight γ3λ2(1− λ) on Qt(St+3,At+3)
...
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Forward View Sarsa(λ)
Written recursively:
Gλt = Rt+1 + γ(1− λ)Qt(St+1,At+1) + γλGλ
t+1
Weight 1 on Rt+1
Weight γ(1− λ) on Qt(St+1,At+1)
Weight γλ on Rt+2
Weight γ2λ(1− λ) on Qt(St+2,At+2)
Weight γ2λ2 on Rt+3
Weight γ3λ2(1− λ) on Qt(St+3,At+3)
...
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Backward View Sarsa(λ)
Just like TD(λ), we use eligibility traces in an online algorithm
But Sarsa(λ) has one eligibility trace for each state-action pair
E0(s, a) = 0
Et(s, a) = (1− αt)γλEt−1(s, a) + αt1(St = s,At = a)
Q(s, a) is updated for every state s and action a
In proportion to TD-error δt and eligibility trace Et(s, a)
δt = Rt+1 + γQt(St+1,At+1)− Q(St ,At)
∀s, a :Q(s, a)← Q(s, a) + δtEt(s, a)
NB: I rolled step size into the traces on this and next slide!
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Backward View Sarsa(λ)
Just like TD(λ), we use eligibility traces in an online algorithm
But Sarsa(λ) has one eligibility trace for each state-action pair
E0(s, a) = 0
Et(s, a) = (1− αt)γλEt−1(s, a) + αt1(St = s,At = a)
Q(s, a) is updated for every state s and action a
In proportion to TD-error δt and eligibility trace Et(s, a)
δt = Rt+1 + γQt(St+1,At+1)− Q(St ,At)
∀s, a :Q(s, a)← Q(s, a) + δtEt(s, a)
NB: I rolled step size into the traces on this and next slide!
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Backward View Sarsa(λ)
Just like TD(λ), we use eligibility traces in an online algorithm
But Sarsa(λ) has one eligibility trace for each state-action pair
E0(s, a) = 0
Et(s, a) = (1− αt)γλEt−1(s, a) + αt1(St = s,At = a)
Q(s, a) is updated for every state s and action a
In proportion to TD-error δt and eligibility trace Et(s, a)
δt = Rt+1 + γQt(St+1,At+1)− Q(St ,At)
∀s, a :Q(s, a)← Q(s, a) + δtEt(s, a)
NB: I rolled step size into the traces on this and next slide!
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Backward View Sarsa(λ)
Just like TD(λ), we use eligibility traces in an online algorithm
But Sarsa(λ) has one eligibility trace for each state-action pair
E0(s, a) = 0
Et(s, a) = (1− αt)γλEt−1(s, a) + αt1(St = s,At = a)
Q(s, a) is updated for every state s and action a
In proportion to TD-error δt and eligibility trace Et(s, a)
δt = Rt+1 + γQt(St+1,At+1)− Q(St ,At)
∀s, a :Q(s, a)← Q(s, a) + δtEt(s, a)
NB: I rolled step size into the traces on this and next slide!
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Backward View Sarsa(λ)
Just like TD(λ), we use eligibility traces in an online algorithm
But Sarsa(λ) has one eligibility trace for each state-action pair
E0(s, a) = 0
Et(s, a) = (1− αt)γλEt−1(s, a) + αt1(St = s,At = a)
Q(s, a) is updated for every state s and action a
In proportion to TD-error δt and eligibility trace Et(s, a)
δt = Rt+1 + γQt(St+1,At+1)− Q(St ,At)
∀s, a :Q(s, a)← Q(s, a) + δtEt(s, a)
NB: I rolled step size into the traces on this and next slide!
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Sarsa(λ) Algorithm
Select A0 according to ⇡0(S0)
Initialize E0(s, a) = 0, for all s, a
For t = 0, 1, 2, . . .
Take action At, observe Rt+1, St+1
Select At+1 according to ⇡t+1(St+1)
�t = Rt+1 + �tQt(St+1, At+1)�Qt(St, At)
Et(St, At) = Et�1(St, At) + ↵t (accumulating trace)
or Et(St, At) = ↵t (replacing trace)
or Et(St, At) = Et�1(St, At) + ↵t(1� Et�1(s, a)) (dutch trace)
For all s, a:
Qt+1(s, a) = Qt(s, a) + �tEt(s, a)
Et(s, a) = �t�tEt�1(s, a)
Lecture 6: Model-Free Control
On-Policy Temporal-Difference Learning
Sarsa(λ)
Sarsa(λ) Gridworld Example
Lecture 6: Model-Free Control
Off-Policy Learning
Outline
1 Introduction
2 Monte-Carlo Control
3 On-Policy Temporal-Difference Learning
4 Off-Policy Learning
5 Summary
Lecture 6: Model-Free Control
Off-Policy Learning
On and Off-Policy Learning
On-policy learning
“Learn on the job”Learn about policy π from experience sampled from π
Off-policy learning
“Look over someone’s shoulder”Learn about policy π from experience sampled from µ
Lecture 6: Model-Free Control
Off-Policy Learning
On and Off-Policy Learning
On-policy learning
“Learn on the job”
Learn about policy π from experience sampled from π
Off-policy learning
“Look over someone’s shoulder”Learn about policy π from experience sampled from µ
Lecture 6: Model-Free Control
Off-Policy Learning
On and Off-Policy Learning
On-policy learning
“Learn on the job”Learn about policy π from experience sampled from π
Off-policy learning
“Look over someone’s shoulder”Learn about policy π from experience sampled from µ
Lecture 6: Model-Free Control
Off-Policy Learning
On and Off-Policy Learning
On-policy learning
“Learn on the job”Learn about policy π from experience sampled from π
Off-policy learning
“Look over someone’s shoulder”Learn about policy π from experience sampled from µ
Lecture 6: Model-Free Control
Off-Policy Learning
On and Off-Policy Learning
On-policy learning
“Learn on the job”Learn about policy π from experience sampled from π
Off-policy learning
“Look over someone’s shoulder”
Learn about policy π from experience sampled from µ
Lecture 6: Model-Free Control
Off-Policy Learning
On and Off-Policy Learning
On-policy learning
“Learn on the job”Learn about policy π from experience sampled from π
Off-policy learning
“Look over someone’s shoulder”Learn about policy π from experience sampled from µ
Lecture 6: Model-Free Control
Off-Policy Learning
Off-Policy Learning
Evaluate target policy π(s, a) to compute V π(s) or Qπ(s, a)
While following behaviour policy µ(s, a)
{S1,A1,R2, ...,ST} ∼ µ
Why is this important?
Learn from observing humans or other agents
Re-use experience generated from old policies π1, π2, ..., πt−1
Learn about optimal policy while following exploratory policy
Learn about multiple policies while following one policy
Lecture 6: Model-Free Control
Off-Policy Learning
Off-Policy Learning
Evaluate target policy π(s, a) to compute V π(s) or Qπ(s, a)
While following behaviour policy µ(s, a)
{S1,A1,R2, ...,ST} ∼ µ
Why is this important?
Learn from observing humans or other agents
Re-use experience generated from old policies π1, π2, ..., πt−1
Learn about optimal policy while following exploratory policy
Learn about multiple policies while following one policy
Lecture 6: Model-Free Control
Off-Policy Learning
Off-Policy Learning
Evaluate target policy π(s, a) to compute V π(s) or Qπ(s, a)
While following behaviour policy µ(s, a)
{S1,A1,R2, ...,ST} ∼ µ
Why is this important?
Learn from observing humans or other agents
Re-use experience generated from old policies π1, π2, ..., πt−1
Learn about optimal policy while following exploratory policy
Learn about multiple policies while following one policy
Lecture 6: Model-Free Control
Off-Policy Learning
Off-Policy Learning
Evaluate target policy π(s, a) to compute V π(s) or Qπ(s, a)
While following behaviour policy µ(s, a)
{S1,A1,R2, ...,ST} ∼ µ
Why is this important?
Learn from observing humans or other agents
Re-use experience generated from old policies π1, π2, ..., πt−1
Learn about optimal policy while following exploratory policy
Learn about multiple policies while following one policy
Lecture 6: Model-Free Control
Off-Policy Learning
Off-Policy Learning
Evaluate target policy π(s, a) to compute V π(s) or Qπ(s, a)
While following behaviour policy µ(s, a)
{S1,A1,R2, ...,ST} ∼ µ
Why is this important?
Learn from observing humans or other agents
Re-use experience generated from old policies π1, π2, ..., πt−1
Learn about optimal policy while following exploratory policy
Learn about multiple policies while following one policy
Lecture 6: Model-Free Control
Off-Policy Learning
Off-Policy Learning
Evaluate target policy π(s, a) to compute V π(s) or Qπ(s, a)
While following behaviour policy µ(s, a)
{S1,A1,R2, ...,ST} ∼ µ
Why is this important?
Learn from observing humans or other agents
Re-use experience generated from old policies π1, π2, ..., πt−1
Learn about optimal policy while following exploratory policy
Learn about multiple policies while following one policy
Lecture 6: Model-Free Control
Off-Policy Learning
Off-Policy Learning
Evaluate target policy π(s, a) to compute V π(s) or Qπ(s, a)
While following behaviour policy µ(s, a)
{S1,A1,R2, ...,ST} ∼ µ
Why is this important?
Learn from observing humans or other agents
Re-use experience generated from old policies π1, π2, ..., πt−1
Learn about optimal policy while following exploratory policy
Learn about multiple policies while following one policy
Lecture 6: Model-Free Control
Off-Policy Learning
Importance Sampling
Importance Sampling
Estimate the expectation of a different distribution
Ex∼d [f (x)] =∑
d(x)f (x)
=∑
d ′(x)d(x)
d ′(x)f (x)
= Ex∼d ′
[d(x)
d ′(x)f (x)
]
Lecture 6: Model-Free Control
Off-Policy Learning
Importance Sampling
Importance Sampling for Off-Policy Monte-Carlo
Use returns generated from µ to evaluate π
Weight return vt according to similarity between policies
Multiply importance sampling corrections along whole episode
Gπ/µt =
π(St ,At)
µ(St ,At)
π(St+1,At+1)
µ(St+1,At+1). . .
π(ST ,AT )
µ(ST ,AT )Gt
Update value towards corrected return
V (St)← V (St) + α(Gπ/µt − V (St)
)
Importance sampling can dramatically increase variance
Lecture 6: Model-Free Control
Off-Policy Learning
Importance Sampling
Importance Sampling for Off-Policy Monte-Carlo
Use returns generated from µ to evaluate π
Weight return vt according to similarity between policies
Multiply importance sampling corrections along whole episode
Gπ/µt =
π(St ,At)
µ(St ,At)
π(St+1,At+1)
µ(St+1,At+1). . .
π(ST ,AT )
µ(ST ,AT )Gt
Update value towards corrected return
V (St)← V (St) + α(Gπ/µt − V (St)
)
Importance sampling can dramatically increase variance
Lecture 6: Model-Free Control
Off-Policy Learning
Importance Sampling
Importance Sampling for Off-Policy Monte-Carlo
Use returns generated from µ to evaluate π
Weight return vt according to similarity between policies
Multiply importance sampling corrections along whole episode
Gπ/µt =
π(St ,At)
µ(St ,At)
π(St+1,At+1)
µ(St+1,At+1). . .
π(ST ,AT )
µ(ST ,AT )Gt
Update value towards corrected return
V (St)← V (St) + α(Gπ/µt − V (St)
)
Importance sampling can dramatically increase variance
Lecture 6: Model-Free Control
Off-Policy Learning
Importance Sampling
Importance Sampling for Off-Policy Monte-Carlo
Use returns generated from µ to evaluate π
Weight return vt according to similarity between policies
Multiply importance sampling corrections along whole episode
Gπ/µt =
π(St ,At)
µ(St ,At)
π(St+1,At+1)
µ(St+1,At+1). . .
π(ST ,AT )
µ(ST ,AT )Gt
Update value towards corrected return
V (St)← V (St) + α(Gπ/µt − V (St)
)
Importance sampling can dramatically increase variance
Lecture 6: Model-Free Control
Off-Policy Learning
Importance Sampling
Importance Sampling for Off-Policy Monte-Carlo
Use returns generated from µ to evaluate π
Weight return vt according to similarity between policies
Multiply importance sampling corrections along whole episode
Gπ/µt =
π(St ,At)
µ(St ,At)
π(St+1,At+1)
µ(St+1,At+1). . .
π(ST ,AT )
µ(ST ,AT )Gt
Update value towards corrected return
V (St)← V (St) + α(Gπ/µt − V (St)
)
Importance sampling can dramatically increase variance
Lecture 6: Model-Free Control
Off-Policy Learning
Importance Sampling
Importance Sampling for Off-Policy TD
Use TD targets generated from µ to evaluate π
Weight TD target r + γV (s ′) by importance sampling
Only need a single importance sampling correction
V (St)← V (St) +
α
(π(St ,At)
µ(St ,At)(Rt+1 + γV (St+1))− V (St)
)
Much lower variance than Monte-Carlo importance sampling
Policies only need to be similar over a single step
Lecture 6: Model-Free Control
Off-Policy Learning
Importance Sampling
Importance Sampling for Off-Policy TD
Use TD targets generated from µ to evaluate π
Weight TD target r + γV (s ′) by importance sampling
Only need a single importance sampling correction
V (St)← V (St) +
α
(π(St ,At)
µ(St ,At)(Rt+1 + γV (St+1))− V (St)
)
Much lower variance than Monte-Carlo importance sampling
Policies only need to be similar over a single step
Lecture 6: Model-Free Control
Off-Policy Learning
Importance Sampling
Importance Sampling for Off-Policy TD
Use TD targets generated from µ to evaluate π
Weight TD target r + γV (s ′) by importance sampling
Only need a single importance sampling correction
V (St)← V (St) +
α
(π(St ,At)
µ(St ,At)(Rt+1 + γV (St+1))− V (St)
)
Much lower variance than Monte-Carlo importance sampling
Policies only need to be similar over a single step
Lecture 6: Model-Free Control
Off-Policy Learning
Importance Sampling
Importance Sampling for Off-Policy TD
Use TD targets generated from µ to evaluate π
Weight TD target r + γV (s ′) by importance sampling
Only need a single importance sampling correction
V (St)← V (St) +
α
(π(St ,At)
µ(St ,At)(Rt+1 + γV (St+1))− V (St)
)
Much lower variance than Monte-Carlo importance sampling
Policies only need to be similar over a single step
Lecture 6: Model-Free Control
Off-Policy Learning
Importance Sampling
Importance Sampling for Off-Policy TD
Use TD targets generated from µ to evaluate π
Weight TD target r + γV (s ′) by importance sampling
Only need a single importance sampling correction
V (St)← V (St) +
α
(π(St ,At)
µ(St ,At)(Rt+1 + γV (St+1))− V (St)
)
Much lower variance than Monte-Carlo importance sampling
Policies only need to be similar over a single step
Lecture 6: Model-Free Control
Off-Policy Learning
Q-Learning
Q-Learning
We now consider off-policy learning of action-values Q(s, a)
No importance sampling is required
Next action may be chosen using behaviour policyAt+1 ∼ µ(St , ·)But we consider probabilities under π(St , ·)Update Q(St ,At) towards value of alternative action
Q(St ,At)← Q(St ,At)+
α
(Rt+1 + γ
∑
a
π(a|St+1)Q(St+1, a)− Q(St ,At)
)
Called Expected Sarsa (when µ = π) or Generalized Q-learning
Lecture 6: Model-Free Control
Off-Policy Learning
Q-Learning
Q-Learning
We now consider off-policy learning of action-values Q(s, a)
No importance sampling is required
Next action may be chosen using behaviour policyAt+1 ∼ µ(St , ·)But we consider probabilities under π(St , ·)Update Q(St ,At) towards value of alternative action
Q(St ,At)← Q(St ,At)+
α
(Rt+1 + γ
∑
a
π(a|St+1)Q(St+1, a)− Q(St ,At)
)
Called Expected Sarsa (when µ = π) or Generalized Q-learning
Lecture 6: Model-Free Control
Off-Policy Learning
Q-Learning
Q-Learning
We now consider off-policy learning of action-values Q(s, a)
No importance sampling is required
Next action may be chosen using behaviour policyAt+1 ∼ µ(St , ·)
But we consider probabilities under π(St , ·)Update Q(St ,At) towards value of alternative action
Q(St ,At)← Q(St ,At)+
α
(Rt+1 + γ
∑
a
π(a|St+1)Q(St+1, a)− Q(St ,At)
)
Called Expected Sarsa (when µ = π) or Generalized Q-learning
Lecture 6: Model-Free Control
Off-Policy Learning
Q-Learning
Q-Learning
We now consider off-policy learning of action-values Q(s, a)
No importance sampling is required
Next action may be chosen using behaviour policyAt+1 ∼ µ(St , ·)But we consider probabilities under π(St , ·)
Update Q(St ,At) towards value of alternative action
Q(St ,At)← Q(St ,At)+
α
(Rt+1 + γ
∑
a
π(a|St+1)Q(St+1, a)− Q(St ,At)
)
Called Expected Sarsa (when µ = π) or Generalized Q-learning
Lecture 6: Model-Free Control
Off-Policy Learning
Q-Learning
Q-Learning
We now consider off-policy learning of action-values Q(s, a)
No importance sampling is required
Next action may be chosen using behaviour policyAt+1 ∼ µ(St , ·)But we consider probabilities under π(St , ·)Update Q(St ,At) towards value of alternative action
Q(St ,At)← Q(St ,At)+
α
(Rt+1 + γ
∑
a
π(a|St+1)Q(St+1, a)− Q(St ,At)
)
Called Expected Sarsa (when µ = π) or Generalized Q-learning
Lecture 6: Model-Free Control
Off-Policy Learning
Q-Learning
Q-Learning
We now consider off-policy learning of action-values Q(s, a)
No importance sampling is required
Next action may be chosen using behaviour policyAt+1 ∼ µ(St , ·)But we consider probabilities under π(St , ·)Update Q(St ,At) towards value of alternative action
Q(St ,At)← Q(St ,At)+
α
(Rt+1 + γ
∑
a
π(a|St+1)Q(St+1, a)− Q(St ,At)
)
Called Expected Sarsa (when µ = π) or Generalized Q-learning
Lecture 6: Model-Free Control
Off-Policy Learning
Q-Learning
Off-Policy Control with Q-Learning
We now allow both behaviour and target policies to improve
The target policy π is greedy w.r.t. Q(s, a)
π(St+1) = argmaxa′
Q(St+1, a′)
The behaviour policy µ is e.g. ε-greedy w.r.t. Q(s, a)
The Q-learning target then simplifies:
Rt+1 + γ∑
a
π(a|St+1)Q(St+1, a)
=Rt+1 + γQ(St+1, argmaxa
Q(St+1, a))
=Rt+1 + γmaxa
Q(St+1, a)
Lecture 6: Model-Free Control
Off-Policy Learning
Q-Learning
Off-Policy Control with Q-Learning
We now allow both behaviour and target policies to improve
The target policy π is greedy w.r.t. Q(s, a)
π(St+1) = argmaxa′
Q(St+1, a′)
The behaviour policy µ is e.g. ε-greedy w.r.t. Q(s, a)
The Q-learning target then simplifies:
Rt+1 + γ∑
a
π(a|St+1)Q(St+1, a)
=Rt+1 + γQ(St+1, argmaxa
Q(St+1, a))
=Rt+1 + γmaxa
Q(St+1, a)
Lecture 6: Model-Free Control
Off-Policy Learning
Q-Learning
Off-Policy Control with Q-Learning
We now allow both behaviour and target policies to improve
The target policy π is greedy w.r.t. Q(s, a)
π(St+1) = argmaxa′
Q(St+1, a′)
The behaviour policy µ is e.g. ε-greedy w.r.t. Q(s, a)
The Q-learning target then simplifies:
Rt+1 + γ∑
a
π(a|St+1)Q(St+1, a)
=Rt+1 + γQ(St+1, argmaxa
Q(St+1, a))
=Rt+1 + γmaxa
Q(St+1, a)
Lecture 6: Model-Free Control
Off-Policy Learning
Q-Learning
Off-Policy Control with Q-Learning
We now allow both behaviour and target policies to improve
The target policy π is greedy w.r.t. Q(s, a)
π(St+1) = argmaxa′
Q(St+1, a′)
The behaviour policy µ is e.g. ε-greedy w.r.t. Q(s, a)
The Q-learning target then simplifies:
Rt+1 + γ∑
a
π(a|St+1)Q(St+1, a)
=Rt+1 + γQ(St+1, argmaxa
Q(St+1, a))
=Rt+1 + γmaxa
Q(St+1, a)
Lecture 6: Model-Free Control
Off-Policy Learning
Q-Learning
Q-Learning Control Algorithm
Theorem
Q-learning control converges to the optimal action-value function,Q(s, a)→ Q∗(s, a), as long as we take each action in each stateinfinitely often.
Lecture 6: Model-Free Control
Off-Policy Learning
Q-Learning
Q-Learning Control Algorithm
Theorem
Q-learning control converges to the optimal action-value function,Q(s, a)→ Q∗(s, a), as long as we take each action in each stateinfinitely often.
Note: no need for greedy behaviour!
Lecture 6: Model-Free Control
Off-Policy Learning
Q-Learning
Q-Learning Algorithm for Off-Policy Control
For t = 0, 1, 2, . . .
Take action At according to ⇡t(St), observe Rt+1, St+1
Qt+1(St, At) = Qt(St, At) + ↵t
⇣Rt+1 + �t max
aQt(St+1, a)�Qt(St, At)
⌘
Lecture 6: Model-Free Control
Off-Policy Learning
Q-Learning
Cliff Walking Example
Lecture 6: Model-Free Control
Summary
Outline
1 Introduction
2 Monte-Carlo Control
3 On-Policy Temporal-Difference Learning
4 Off-Policy Learning
5 Summary
Lecture 6: Model-Free Control
Summary
Relationship Between DP and TD
Full Backup (DP) Sample Backup (TD)
Bellman Expectation
s
a
V!(s)
s'
r
V!(s')
Equation for V π(s) Iterative Policy Evaluation TD Learning
Bellman Expectation
s,a
s'
Q!(s,a)
r
a' Q!(s',a')
s,a
r
a'
s'
Equation for Qπ(s, a) Q-Policy Iteration Sarsa
Bellman Optimality
s,a
s'
Q*(s,a)
r
a' Q*(s',a')
Equation for Q∗(s, a) Q-Value Iteration Q-Learning
Lecture 6: Model-Free Control
Summary
Relationship Between DP and TD (2)
Full Backup (DP) Sample Backup (TD)
Iterative Policy Evaluation TD Learning
V (s)← E [r + γV (s ′) | s] V (s)α← r + γV (s ′)
Q-Policy Iteration Sarsa
Q(s, a)← E [r + γQ(s ′, a′) | s, a] Q(s, a)α← r + γQ(s ′, a′)
Q-Value Iteration Q-Learning
Q(s, a)← E[r + γ max
a′∈AQ(s ′, a′) | s, a
]Q(s, a)
α← r + γ maxa′∈A
Q(s ′, a′)
where xα← y ≡ x ← x + α(y − x)
Lecture 6: Model-Free Control
Summary
Q-learning overestimates
Q-learning has a problem
Recall
maxa
Qt(St+1, a) = Qt(St+1, argmaxa
Qt(St+1, a))
Q-learning uses same values to select and to evaluate
... but values are approximate
Therefore:
more likely to select overestimated valuesless likely to select underestimated values
This causes upward bias
Lecture 6: Model-Free Control
Summary
Q-learning overestimates
Q-learning has a problem
Recall
maxa
Qt(St+1, a) = Qt(St+1, argmaxa
Qt(St+1, a))
Q-learning uses same values to select and to evaluate
... but values are approximate
Therefore:
more likely to select overestimated valuesless likely to select underestimated values
This causes upward bias
Lecture 6: Model-Free Control
Summary
Q-learning overestimates
Q-learning has a problem
Recall
maxa
Qt(St+1, a) = Qt(St+1, argmaxa
Qt(St+1, a))
Q-learning uses same values to select and to evaluate
... but values are approximate
Therefore:
more likely to select overestimated valuesless likely to select underestimated values
This causes upward bias
Lecture 6: Model-Free Control
Summary
Q-learning overestimates
Q-learning has a problem
Recall
maxa
Qt(St+1, a) = Qt(St+1, argmaxa
Qt(St+1, a))
Q-learning uses same values to select and to evaluate
... but values are approximate
Therefore:
more likely to select overestimated valuesless likely to select underestimated values
This causes upward bias
Lecture 6: Model-Free Control
Summary
Q-learning overestimates
Q-learning has a problem
Recall
maxa
Qt(St+1, a) = Qt(St+1, argmaxa
Qt(St+1, a))
Q-learning uses same values to select and to evaluate
... but values are approximate
Therefore:
more likely to select overestimated valuesless likely to select underestimated values
This causes upward bias
Lecture 6: Model-Free Control
Summary
Q-learning overestimates
Q-learning has a problem
Recall
maxa
Qt(St+1, a) = Qt(St+1, argmaxa
Qt(St+1, a))
Q-learning uses same values to select and to evaluate
... but values are approximate
Therefore:
more likely to select overestimated values
less likely to select underestimated values
This causes upward bias
Lecture 6: Model-Free Control
Summary
Q-learning overestimates
Q-learning has a problem
Recall
maxa
Qt(St+1, a) = Qt(St+1, argmaxa
Qt(St+1, a))
Q-learning uses same values to select and to evaluate
... but values are approximate
Therefore:
more likely to select overestimated valuesless likely to select underestimated values
This causes upward bias
Lecture 6: Model-Free Control
Summary
Q-learning overestimates
Q-learning has a problem
Recall
maxa
Qt(St+1, a) = Qt(St+1, argmaxa
Qt(St+1, a))
Q-learning uses same values to select and to evaluate
... but values are approximate
Therefore:
more likely to select overestimated valuesless likely to select underestimated values
This causes upward bias
Lecture 6: Model-Free Control
Summary
Q-learning overestimates
Lecture 6: Model-Free Control
Summary
Double Q-learning
Q-learning uses same values to select and to evaluate
Rt+1 + γQt(St+1, argmaxa
Qt(St+1, a))
Solution: decouple selection from evaluation
Double Q-learning:
Rt+1 + γQ ′t(St+1, argmaxa
Qt(St+1, a))
Rt+1 + γQt(St+1, argmaxa
Q ′t(St+1, a))
Then update one at random for each experience
Lecture 6: Model-Free Control
Summary
Double Q-learning
Q-learning uses same values to select and to evaluate
Rt+1 + γQt(St+1, argmaxa
Qt(St+1, a))
Solution: decouple selection from evaluation
Double Q-learning:
Rt+1 + γQ ′t(St+1, argmaxa
Qt(St+1, a))
Rt+1 + γQt(St+1, argmaxa
Q ′t(St+1, a))
Then update one at random for each experience
Lecture 6: Model-Free Control
Summary
Double Q-learning
Q-learning uses same values to select and to evaluate
Rt+1 + γQt(St+1, argmaxa
Qt(St+1, a))
Solution: decouple selection from evaluation
Double Q-learning:
Rt+1 + γQ ′t(St+1, argmaxa
Qt(St+1, a))
Rt+1 + γQt(St+1, argmaxa
Q ′t(St+1, a))
Then update one at random for each experience
Lecture 6: Model-Free Control
Summary
Double Q-learning
Q-learning uses same values to select and to evaluate
Rt+1 + γQt(St+1, argmaxa
Qt(St+1, a))
Solution: decouple selection from evaluation
Double Q-learning:
Rt+1 + γQ ′t(St+1, argmaxa
Qt(St+1, a))
Rt+1 + γQt(St+1, argmaxa
Q ′t(St+1, a))
Then update one at random for each experience
Lecture 6: Model-Free Control
Summary
Double Q-learning
Lecture 6: Model-Free Control
Summary
Double DQN on Atari
DQN
0 100
200
400
800
1600
3200
6400
Normalized score
Hum
an
Double DunkWizard of Wor
Video PinballGravitar
Private EyeAsterix
AsteroidsMontezuma's Revenge
FrostbiteVenture
Ms. PacmanZaxxonBowling
SeaquestAlien
AmidarTutankhamCentipede
Bank HeistRiver Raid
Up and DownChopper Command
Q*BertH.E.R.O.
Battle ZoneKung-Fu Master
Beam RiderIce Hockey
EnduroName This Game
Fishing DerbyJames Bond
KangarooSpace Invaders
Time PilotFreeway
PongGopherTennis
Road RunnerCrazy Climber
BoxingKrull
Star GunnerDemon Attack
AtlantisAssault
RobotankBreakout
Lecture 6: Model-Free Control
Summary
Double DQN on Atari
DQNDouble DQN
0 100
200
400
800
1600
3200
6400
Normalized score
Hum
an
Double DunkWizard of Wor
Video PinballGravitar
Private EyeAsterix
AsteroidsMontezuma's Revenge
FrostbiteVenture
Ms. PacmanZaxxonBowling
SeaquestAlien
AmidarTutankhamCentipede
Bank HeistRiver Raid
Up and DownChopper Command
Q*BertH.E.R.O.
Battle ZoneKung-Fu Master
Beam RiderIce Hockey
EnduroName This Game
Fishing DerbyJames Bond
KangarooSpace Invaders
Time PilotFreeway
PongGopherTennis
Road RunnerCrazy Climber
BoxingKrull
Star GunnerDemon Attack
AtlantisAssault
RobotankBreakout
Lecture 6: Model-Free Control
Summary
Questions?